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Receiver Design for OTFS with Fractionally Spaced Sampling Approach

Yao Ge, Qinwen Deng, P. C. Ching, Zhi Ding

arXiv:2009.00806v2cs.ITeess.SPmath.IT

TL;DR

Practical OTFS receiver design must move beyond ideal pulses and on-grid delay/Doppler assumptions. This paper derives a rectangular-pulse channel model and develops FSS-based message-passing receivers; the authors report stronger performance than existing solutions and robustness to CSI uncertainty.

  • Problem

    Many OTFS studies rely on ideal bi-orthogonal pulses, which are unrealizable, while practical rectangular-pulse receiver designs have used impractical on-grid delay or Doppler assumptions.

    Source: §I · p. 2

  • Method

    The paper derives a delay-Doppler input-output relationship for rectangular-pulse OTFS and proposes FSS-based ICMP and TMP receivers that exploit channel sparsity and frequency diversity.

    Source: §I · pp. 3–4

  • Results

    The proposed FSS receivers achieve stronger performance than existing solutions, and ICMP and TMP are robust to CSI uncertainty.

    Source: §I · p. 4

  • Takeaways & Limitations

    FSS-based message-passing receivers provide a practical OTFS approach using rectangular pulses without requiring on-grid delay or Doppler shifts.

    Source: §I · pp. 3–4

Abstract

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The recent emergence of orthogonal time frequency space (OTFS) modulation as a novel PHY-layer mechanism is more suitable in high-mobility wireless communication scenarios than traditional orthogonal frequency division multiplexing (OFDM). Although multiple studies have analyzed OTFS performance using theoretical and ideal baseband pulseshapes, a challenging and open problem is the development of effective receivers for practical OTFS systems that must rely on non-ideal pulseshapes for transmission. This work focuses on the design of practical receivers for OTFS. We consider a fractionally spaced sampling (FSS) receiver in which the sampling rate is an integer multiple of the symbol rate. For rectangular pulses used in OTFS transmission, we derive a general channel input-output relationship of OTFS in delay-Doppler domain without the common reliance on impractical assumptions such as ideal bi-orthogonal pulses and on-the-grid delay/Doppler shifts. We propose two equalization algorithms: iterative combining message passing (ICMP) and turbo message passing (TMP) for symbol detection by exploiting delay-Doppler channel sparsity and the frequency diversity gain via FSS. We analyze the convergence performance of TMP receiver and propose simplified message passing (MP) receivers to further reduce complexity. Our FSS receivers demonstrate stronger performance than traditional receivers and robustness to the imperfect channel state information knowledge.

I. INTRODUCTION

The introduction motivates practical OTFS receivers for high-mobility channels, where ideal pulses and on-grid delay/Doppler assumptions are unsuitable. It proposes fractionally spaced sampling and message-passing receivers for rectangular-pulse OTFS, reporting stronger performance than existing solutions and robustness to CSI uncertainty.

I. INTRODUCTION

  • OTFS places symbols in the delay-Doppler domain, transforming a rapidly varying time-frequency channel into a near-stationary one that can simplify receiver design and channel estimation.The passage contrasts this with OFDM's performance degradation in fast time-varying channels due to loss of subcarrier orthogonality and inter-carrier interference.Source: §I · p. 2
  • Earlier rectangular-pulse OTFS receivers using one CP per frame still assumed delay or Doppler shifts fell on a predetermined sampling grid.The paper identifies this assumption as impractical for real OTFS deployment.Source: §I · p. 3
  • The paper derives a delay-Doppler input-output relationship for rectangular pulses without ideal bi-orthogonal pulses or on-grid delay/Doppler shifts.The authors note that practical cases entail ISI and extraneous phase shifts.Source: §I · p. 3
  • FSS-based ICMP and TMP exploit delay-Doppler channel sparsity and frequency diversity for OTFS symbol detection.The receiver structure uses fractionally spaced sampling, with the two algorithms presented as moderate-complexity detectors.Source: §I · pp. 3–4
  • A simplified message-passing algorithm truncates weak factor-graph edges to reduce complexity without significant performance loss.The paper also analyzes TMP performance and convergence using an EXIT chart.Source: §I · p. 4
  • The proposed FSS OTFS receivers achieve stronger performance than existing solutions, while ICMP and TMP are robust to CSI uncertainty.The introduction identifies both receiver performance and channel-state-information robustness as outcomes.Source: §I · p. 4

A. Basic Concepts of OTFS

  • Ideal bi-orthogonal pulses eliminate cross-symbol interference, but cannot be realized in practice; practical rectangular-pulse OTFS therefore requires receiver equalization.The basic-concepts passage introduces transmit and receive pulses and their bi-orthogonal condition.Source: §I · p. 2, A. Basic Concepts of OTFS · p. 5

B. OTFS System Model

OTFS maps information symbols from the delay-Doppler plane to the time-frequency plane and then to a transmitted time-domain signal. The system model describes the multipath channel and the receiver transformations used to recover delay-Doppler symbols.

B. OTFS System Model

  • Information symbols on the delay-Doppler plane are mapped to the time-frequency plane by ISFFT before transmission.The Heisenberg transform then converts the time-frequency signal into a time-domain signal.Source: B. OTFS System Model · pp. 5–6
  • A cyclic prefix is inserted to address inter-frame interference, and the channel model represents L multipath components with gains, delays, and Doppler shifts.The sampling interval is T_s = 1/(M∆f); the Doppler description also includes fractional shifts and the sampled equivalent filter response.Source: B. OTFS System Model · p. 6
  • At the receiver, the filtered signal is transformed to the time-frequency domain by the Wigner transform and then sampled before SFFT recovers delay-Doppler symbols.The receiver model includes additive white Gaussian noise.Source: B. OTFS System Model · pp. 6–7
  • The section characterizes OTFS input-output relationships in both the time-frequency and delay-Doppler domains.The text presents these relationships as the basis for modeling OTFS with specific pulses.Source: B. OTFS System Model · p. 7

III. OTFS MODEL BASED ON SSS FOR RECTANGULAR PULSES

The section derives a delay-Doppler input-output model for rectangular-pulse OTFS without requiring ideal bi-orthogonality or on-grid Doppler shifts. The model captures interference, including under the on-grid simplification.

III. OTFS MODEL BASED ON SSS FOR RECTANGULAR PULSES

  • With time-varying channels, rectangular pulses cannot satisfy the ideal bi-orthogonal condition, so receiver ISI is not eliminated.The text states that ideal pulses meeting that condition cannot be realized in practice and that equalization is needed for satisfactory reception.Source: §III · p. 8
  • The delay-Doppler relationship includes desired signal, ICI, and ISI terms, while the Doppler response peaks at q = 0 and decreases rapidly as |q| grows.This decay supports approximating the response with a limited set of significant q values.Source: §III · pp. 9–10
  • Theorem 3 gives the delay-Doppler input-output relationship for OTFS with rectangular pulses, including the effects of fractional Doppler shifts.The model’s received signal includes contributions from channel paths and a noise term.Source: §III · p. 9
  • Even when Doppler shifts are exactly on-grid, the simplified relationship retains ISI, so practical rectangular-pulse OTFS still requires receiver equalization.The section also notes that existing models often rely on impractical ideal-pulse or on-grid assumptions.Source: §III · pp. 8, 10

IV. RECEIVER DESIGN FOR OTFS WITH FSS

The receiver design uses fractionally spaced sampling (FSS) to retain signal information that symbol-spaced sampling may miss with bandlimiting filters. Its sampling branches form a sparse SIMO model, and the proposed G = 2 setting does not require on-grid delay or Doppler shifts.

IV. RECEIVER DESIGN FOR OTFS WITH FSS

  • For RRC transmit and receive filters, symbol-spaced sampling may not preserve sufficient statistics because the filter bandwidth can exceed the minimum required bandwidth.The text also notes possible sensitivity to sampling instants.Source: §IV · p. 10
  • The FSS receiver is designed for rectangular pulses and RRC filters, and the approach can be generalized to non-rectangular pulses.The stated motivation is to admit sufficient statistics and improve equalization performance.Source: §IV · p. 11

A. Receiver Structure

  • Sampling at an integer multiple G of the symbol rate makes the FSS receiver equivalent to G parallel channels with correlated noise.The receive-filter output is represented in polyphase form after cyclic-prefix removal.Source: A. Receiver Structure · p. 11
  • The per-branch OTFS input-output relationships can be vectorized as y_g ≃ H_g x + z_g, with sparse H_g when the number of nonzero channel terms D is much smaller than NM.The model uses the multiple receptions for diversity combining.Source: A. Receiver Structure · pp. 11–12
  • The design focuses on G = 2 because the RRC-filter bandwidth makes this interval sufficient to preserve the sufficient signal statistic.The receivers can potentially exploit channel spectrum diversity without multiple antennas or RF chains, and do not require on-grid delay or Doppler shifts.Source: A. Receiver Structure · p. 12

B. ICMP Receiver Equalization

The ICMP receiver turns OTFS symbol detection into message passing on a sparse factor graph, avoiding direct large-scale detection. It combines the two receptions and uses iterative updates, though dense short-cycle graphs and inaccurate interference approximations can impair performance.

B. ICMP Receiver Equalization

  • ICMP combines the two receptions and passes messages iteratively between observation nodes and connected variable nodes.Interference components are approximated as Gaussian, and channel sparsity supports approximate independence of observation components for a given symbol.Source: B. ICMP Receiver Equalization · pp. 12–13
  • Direct joint MAP detection is intractable at large NM, while matrix inversion is computationally demanding, motivating a lower-complexity detector.The transmitted symbols and observations form a sparsely connected factor graph based on nonzero entries of H.Source: B. ICMP Receiver Equalization · pp. 12–13
  • The receiver updates symbol probabilities only when the current iteration improves and stops when η(κ) = 1 or the iteration limit is reached.A damping factor is used to control convergence speed.Source: B. ICMP Receiver Equalization · p. 15
  • ICMP performance may degrade with dense short-cycle graphs or when Gaussian interference approximations are inaccurate.Off-grid channel delays and Doppler shifts often produce high-density OTFS graphs.Source: B. ICMP Receiver Equalization · p. 15

C. TMP Receiver Equalization

TMP lets two MP equalizers cooperate by exchanging soft information. Each equalizer uses the other’s extrinsic LLRs as a priori information, and the exchange continues for a set number of iterations.

C. TMP Receiver Equalization

  • TMP connects two MP equalizers by passing each equalizer’s extrinsic LLRs to the other as a priori information.The soft-information exchange forms an iterative loop in which more reliable information from one equalizer helps the other.Source: C. TMP Receiver Equalization · p. 16
  • TMP uses modest modifications to the MP equalizers’ inputs and outputs to incorporate a priori and a posteriori information.The first equalizer’s a posteriori LLRs contribute to the extrinsic information sent to the second equalizer, which returns extrinsic LLRs to the first.Source: C. TMP Receiver Equalization · p. 16
  • The equalizers pass only extrinsic information because repeated exchange of correlated messages would reduce iterative efficiency and cause performance loss.The receiver terminates after a specified number of iterations.Source: C. TMP Receiver Equalization · pp. 16–17

D. Performance Analysis of TMP Receiver

The TMP analysis uses EXIT charts to track mutual-information transfer and iterative trajectories. In the reported QPSK example, trajectories reach the equalizers’ convergence point, and performance gains become negligible beyond three iterations.

D. Performance Analysis of TMP Receiver

V. REDUCED COMPLEXITY RECEIVERS

The simplified MP receivers reduce complexity by trimming factor-graph edges, retaining the largest channel terms and approximating the rest as Gaussian. Edge trimming trades possible performance loss for lower complexity and may also reduce short cycles.

V. REDUCED COMPLEXITY RECEIVERS

  • Edge trimming reduces the number of message-passing updates and can also reduce short cycles, which may improve performance despite possible approximation loss.The number of nonzero channel ISI terms D can exceed 150 in the reported experiments because of off-grid delays and Doppler shifts.Source: §V · p. 19
  • The simplified MP algorithm retains the R largest channel-coefficient connections for each observation and approximates the remaining interference as Gaussian.The removed edges correspond to the remaining D − R terms.Source: §V · pp. 19–20
  • Simplified ICMP and TMP complexity scales with MNGR(11Q + 1) and n_t n_iter MNGR(11Q + 1), respectively.The corresponding unsimplified receiver complexities are n_iter MNGD(11Q + 1) and n_t n_iter MNGD(11Q + 1).Source: §V · pp. 19–20

VI. SIMULATION RESULTS

Across simulations, FSS receivers outperform SSS benchmarks, with TMP ahead of ICMP; performance also varies with approximation order, user velocity, system parameters, and CSI uncertainty.

VI. SIMULATION RESULTS

  • Increasing E from 0 to 6 significantly improves BER, but performance then saturates as complexity rises.The experiments use E = 6 unless otherwise stated; many small ISI taps from off-grid shifts limit further gains.Source: Figure 6 · p. 21, §VI · p. 21
  • FSS receivers significantly outperform SSS receivers, with only a modest BER difference between on-grid and off-grid delay/Doppler shifts.The comparison attributes the FSS gain to use of channel diversity and reports robustness to off-grid shifts.Source: Figure 7 · p. 22, §VI · p. 22
  • TMP outperforms ICMP through turbo iterations, while ICMP is more prone to convergence to a local optimum.The passage attributes this difference to short cycles in ICMP’s channel factor graph.Source: §VI · p. 22
  • BER improves as user velocity rises from 5 to 500 km/h and saturates beyond 500 km/h.The passage relates this pattern to resolving more paths in the Doppler dimension and gaining diversity.Source: §VI · p. 22, Figure 8 · p. 23
  • As M and N decrease, ICMP and TMP performance degrades; FSS still provides gains with higher-order modulation such as 16QAM.The passage attributes degradation to lower delay-Doppler grid resolution and loss of resolvable paths.Source: §VI · p. 23, Figure 9 · p. 24
  • With R = 50, simplified MP receivers reduce complexity by about a factor of 3, with insignificant performance loss at smaller R.As R increases, S-ICMP and S-TMP approach the performance of ICMP and TMP, respectively.Source: §VI · p. 23, Figure 10 · p. 24
  • Modest CSI uncertainty causes mild performance loss, without a sudden large drop as uncertainty grows.The authors characterize the proposed receiver architecture as robust to typical CSI errors.Source: Figure 11 · p. 24, §VI · p. 25

VII. CONCLUSION

The paper derives a delay-Doppler channel model for practical rectangular pulses and develops FSS receivers that exploit channel diversity. Its simulations show stronger performance than conventional receivers and robustness to channel uncertainty and modeling errors.

VII. CONCLUSION

  • The derived delay-Doppler input-output relationship for rectangular pulses yields a compact, sparse channel representation.The work also identifies transmitter bandlimiting and receiver matched filtering as required with rectangular OTFS pulses.Source: §VII · p. 25
  • The RRC-based FSS framework supports two delay-Doppler symbol detectors that exploit channel diversity, and TMP shows rapid convergence in EXIT-chart analysis.Source: §VII · p. 25
  • Simplified MP reduces complexity for both proposed receivers, while results show stronger performance than conventional receivers and robustness to channel uncertainty and modeling errors.Source: §VII · p. 25

APPENDIX A

A change of variable is used to rewrite equation (49), completing the proof.

APPENDIX A

  • Changing variables with t′′ = t′ − pTs − n′T rewrites equation (49) and completes the proof.Source: Appendix A · p. 26

APPENDIX C

Appendix C combines prior expressions to derive the delay-Doppler input-output relationship, including a sampled cross-ambiguity function.

APPENDIX C

  • Combining equations (12), (13), and (16) derives the delay-Doppler input-output relationship for n′ = n and n′ = [n − 1]N.Source: Appendix C · p. 26
  • Substituting equation (16) into equation (61) yields an expression using the sampled cross-ambiguity function.Source: Appendix C · pp. 27–28
  • Combining equations (66) and (72) yields the delay-Doppler input-output relationship in equation (18), completing the proof.Source: Appendix C · p. 29
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